ABC and Mochizuki

Apparently we are living through some of the greatest times in mathematics, although the reasons are obscure to anyone who is not a professional mathematician. The last thing that most of us pretended to understand happened almost two decades ago, when Andrew Wiles proved Fermat’s last theorem (which states that an+bn=cn has no integer solutions if n>2). Since this August there has been immense excitement, as well as skepticism, about a proof of something called the ABC Conjecture by the well-known Japanese mathematician Shin-Ichi Mochizuki. Does Shin-Ichi really mean “the new one”?

I would have preferred not to comment seriously on Mochizuki’s work before much more thought had gone into the very basics, but judging from the internet activity, there appears to be much interest in this subject, especially from young people. It would obviously be very nice if they were to engage with this circle of ideas, regardless of the eventual status of the main result of interest. That is to say, the current sense of urgency to understand something seems generally a good thing. So I thought I’d give the flimsiest bit of introduction imaginable at this stage.
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For anyone who wants to really get going, I recommend as starting point some familiarity with two papers, ‘The Hodge-Arakelov theory of elliptic curves (HAT)’ and ‘The Galois-theoretic Kodaira-Spencer morphism of an elliptic curve (GTKS).’

Beyond this point I’m not sure I can even pronounce thing correctly. One is a little reassured when a professional mathematician blogs in Quomodocumque:

I have not even begun to understand Shin’s approach to the conjecture. But it’s clear that it involves ideas which are completely outside the mainstream of the subject. Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space.

This blog is merely meant to point to an explanation of the ABC Conjecture from Nature:

The abc conjecture, proposed independently by David Masser and Joseph Oesterle in 1985, might not be as familiar to the wider world as Fermat’s Last Theorem, but in some ways it is more significant. “The abc conjecture, if proved true, at one stroke solves many famous Diophantine problems, including Fermat’s Last Theorem,” says Dorian Goldfeld, a mathematician at Columbia University in New York. “If Mochizuki’s proof is correct, it will be one of the most astounding achievements of mathematics of the twenty-first century.”

Like Fermat’s theorem, the abc conjecture refers to equations of the form a+b=c. It involves the concept of a square-free number: one that cannot be divided by the square of any number. Fifteen and 17 are square free-numbers, but 16 and 18 — being divisible by 42 and 32, respectively — are not.

The ‘square-free’ part of a number n, sqp(n), is the largest square-free number that can be formed by multiplying the factors of n that are prime numbers. For instance, sqp(18)=2×3=6.

If you’ve got that, then you should get the abc conjecture. It concerns a property of the product of the three integers axbxc, or abc — or more specifically, of the square-free part of this product, which involves their distinct prime factors. It states that for integers a+b=c, the ratio of sqp(abc)r/c always has some minimum value greater than zero for any value of r greater than 1. For example, if a=3 and b=125, so that c=128, then sqp(abc)=30 and sqp(abc)2/c = 900/128. In this case, in which r=2, sqp(abc)r/c is nearly always greater than 1, and always greater than zero.
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[Mochizuki] has developed techniques that very few other mathematicians fully understand and that invoke new mathematical ‘objects’ — abstract entities analogous to more familiar examples such as geometric objects, sets, permutations, topologies and matrices. “At this point, he is probably the only one that knows it all,” says Goldfeld.

Conrad says that the work “uses a huge number of insights that are going to take a long time to be digested by the community”. The proof is spread across four long papers, each of which rests on earlier long papers. “It can require a huge investment of time to understand a long and sophisticated proof, so the willingness by others to do this rests not only on the importance of the announcement but also on the track record of the authors,” Conrad explains.

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For other recent mathematical tours de force — the proofs of Fermat’s last theorem, by Andrew Wiles at Princeton in 1995, and the Poincaré conjecture, by Grigory Perelman, a Russian mathematician, in 2003 — other experts could not immediately tell whether the proofs were valid, but “at least in some outline version, they understood how this approach made sense,” said Nets Katz, a mathematician at Indiana University

For Dr. Mochizuki’s abc conjecture proof, “that seems to be completely missing, and I’ve never seen that in my life,” Dr. Katz said. “It just seems a little odd that most of the people who say positive things about it cannot say what are the ingredients of the proof.” ”